The Brazilian, 30, was quoted as saying he could sign for United this month by a Fox Sports Brazil journalist.
Asked if he had agreed to join United, Dante supposedly replied: "Not yet, I am still with Bayern. I am flattered by Manchester United's interest and will decide by the end of the month."
Bayern Munich treble winners
Dante has over two years left to run on his Bayern contract, and would become the first 30-something United have bought since Edwin van der Sar in 2005.
An ever-present in central defence during Bayern's treble-winning campaign last season, Dante is vying with Chelsea's David Luíz to partner Thiago Silva at the World Cup.
UPDATE: Marcus Marin, Dante's Germany-based agent, insists Dante is actually in discussions with Bayern over a new contract and has ruled out any chance of a move this month.
"There is no way that is going to happen," he told Press Association Sport. "He is not going to leave Bayern in the window.
"At the moment, Bayern are the only people we have spoken to anyway because we are talking over a new contract.
"We are still waiting for an answer from them and, although things happen very quickly in football, until that situation changes, we cannot speak to anyone else anyway."